This partition lattice is then isomorphic to the lattice of the positive integers ordered by divisibility; the supernorm map \(\widehat{N}:\mathcal {P}\rightarrow \mathbb {N}\) , which is clearly reversible, gives the isomorphism explicitly.The second author presented a preliminary overview of these ideas and Fig. REF  at [1]}.
Morally speaking, the lattice bijection of Figure REF  unifies the map of Alladi-Erdős on partitions into prime parts [2]}, the lattice-theoretic backdrop of Andrews' theory of partition ideals [3]}, and aspects of the additive-multiplicative theory in [4]}.Table REF  lists some of the isomorphism relations explicitly.
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